Fractional advection‐dispersion equation: A classical mass balance with convolution‐Fickian Flux

By J. H. Cushman, T. Ginn. 2000

175 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Environmental Engineering, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/dd443b91607fb83ab34d40b2be77f48c56efdcc4

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In Civil and Structural Engineering

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Citations in the corpus

Finite difference methods for two-dimensional fractional dispersion equation (2006, background): “Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate 0021-9991/$ - see front matter 2005 Elsevier Inc.”
Fractional conservation of mass (2008, background): “The fractional advection dispersion equation is known to be a special case of a general transport equation with convolution flux [7] and a limit case of the continuous time random walk with power-law particle jumps [4,10].”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, background): “Fractional derivatives are non-local operators of convolution type [1,6,20].”
TEMPERED FRACTIONAL CALCULUS (2015, background): “They are now used to model a wide variety of problems in physics [24, 35,38,50,51,55,65], finance [23,27,37,42,60,61], biology [2,4,22,26,36], and hydrology [1,7,8,15,17,62].”
Possible problems of scale dependency in applications of the three‐dimensional fractional advection‐dispersion equation to natural porous media (2002, background): “This causes the resulting mass transport (Levy diffusion) at any point x to be nonlocal, i.e. mass located at a finite distance from x can cause a flux at x [Cushman and Ginn, 2000; Molz et al., 2002].”
Physically based modeling in catchment hydrology at 50: Survey and outlook (2015, background): “More recently fractional PDEs have been proposed to address anomalous dispersion in solute transport for surface and subsurface applications [Benson et al., 2000; Cushman and Ginn, 2000; Zhang et al., 2005; Deng et al., 2006].”
Fractional advection‐dispersion equations for modeling transport at the Earth surface (2009, methodology): “The space-time fractional diffusion equation is a special case of this stochastic model, in which the convolution kernel is a power law [ Cushman and Ginn, 1993, 2000 ].”
Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications (2009, background): “, space nonlocality) [39,106,127,148], and it also might depend on the temporal history of concentration ‘‘loading” at that location (time-nonlocality) [36,37,48,49,111,128].”
Perspective on theories of non-Fickian transport in heterogeneous media (2009, result): “This is consistent with the observation [19] that the FF-ADE and FD-ADE, whose kernels are identical when V is constant, coincide with the one-dimensional version of the stnL (13) and, thus, the reduced one-dimensional version of the stnADE (12), as discussed in Section 3.”
Modeling non‐Fickian transport in geological formations as a continuous time random walk (2006, background): “FDEs have also been demonstrated to be special cases of other transport formulations [e.g., Cushman and Ginn, 2000].”
A finite element solution for the fractional advection–dispersion equation (2008, methodology): “By the same token, FADE with constant coefficients may be obtained from Neuman’s theory upon taking the underlying random velocity field to be stationary, in which case Neuman’s theory reduces to that of Cushman and Ginn [10].”
Finite di$erence approximations for fractional advection-dispersion &ow equations (2004, methodology): “Keywords: Finite di$erence approximation; Stability; Backward Euler method; Implicit Euler method; Radial dispersion; Radial advection; Fractional di$usion; Fractional derivative; Fractional advection–dispersion; Numerical fractional ADE”